C₁₃₁⋊(D₄₉⋊C₃) #50310034

order	38514
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	C₁₃₁⋊(D₄₉⋊C₃)
unambiguous semidirect product	(C₄₉⋊C₃)⋊D₁₃₁
unambiguous semidirect product	D₆₄₁₉⋊C₃

Splittings

Fitting, derived, Hall	C₄₉×C₁₃₁ ⋊ C₂×C₃
Hall	C₁₃₁×C₄₉⋊C₃ ⋊ C₂
Hall	D₆₄₁₉ ⋊ C₃
Hall	C₄₉⋊C₃ ⋊ D₁₃₁
Sylow	C₁₃₁ ⋊ D₄₉⋊C₃
Sylow	C₄₉ ⋊ C₃×D₁₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₉×C₁₃₁	C₂×C₃
Frattini	C₇	ΓL(1,𝔽₈)⋊D₁₃₁
socle	C₇×C₁₃₁	AGL(1,𝔽₇)
LCS	G ▷ C₄₉×C₁₃₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 180
transitive degree	μ_t(G) = 6419

A minimal faithful permutation representation — 2 generators on 180 points:

(2,10,12,4,5,13)(3,22,24,8,9,25)(6,34,21,15,16,43)(7,39,40,19,20,49)(11,18,17,23,28,36)(14,37,46,32,33,47)(26,38,41,44,27,30)(29,42,31,45,35,48)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)(177,178)(179,180);(1,23,20,31,30,35,4,8,45,14,13,40,15,19,22,46,5,34,36,28,43,24,44,38,37,33,27,26,25,21,18,17,16,10,47,9,7,6,49,12,32,29,3,2,42,41,48,39,11)(50,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131,133,135,137,139,141,143,145,147,149,151,153,155,157,159,161,163,165,167,169,171,173,175,177,179,180,178,176,174,172,170,168,166,164,162,160,158,156,154,152,150,148,146,144,142,140,138,136,134,132,130,128,126,124,122,120,118,116,114,112,110,108,106,104,102,100,98,96,94,92,90,88,86,84,82,80,78,76,74,72,70,68,66,64,62,60,58,56,54,52)

Polycyclic presentation:

2,3,7,131,7|c^7=e^6,c^a=c^6e,d^a=d^130,e^a=e^6,c^b=c^2e^3,e^b=e^2

Conjugacy classes & character table

1249 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38514, 12);

C131⋊(D49⋊C3) #50310034